In this project the fundamentals of the asymmetric elasticity theory are used to consider one- and two-dimensional static boundary-value problems. The exact analytical solution of each problem is compared with the corresponding solution obtained in the framework of the classical theory of elasticity. The comparison is made in terms of macro-parameters introduced to characterize the degree of difference between these solutions. The analysis of the obtained results shows that for each problem under consideration this difference is not essential. It is worthy of note that the macro-parameters used for comparison can be constructively measured by experiment. The obtained results can be used to outline a key diagram of experiments enabling one to detect the effects of "couple" response of the examined medium.

Problems of material deformation, in which deformation of a medium is described not only by the displacement vector but also by the rotation vector, have long been in the focus of scientists' interest. A medium simulated in such a way is commonly referred to as the Cosserat continuum and the theory describing its behavior is generally known in the literature as the couple-stress, asymmetric, or microstructure theory of elasticity.

In the framework of the Cosserat continuum theory the displacements of particles in the examined medium are described in terms of two variables - an ordinary displacement field and kinematically independent vector field, which is introduced to characterize small rotations of particles. Thus, in the couple-stress theory there are two independent kinematic unknown quantities, and the stress tensor and the couple-stress tensor are asymmetric. In the context of this theory the elastic behavior of isotropic linear medium is described by six elastic constants: two Lame constants and four new constants describing microstructure. In the case of quadratic-nonlinear medium the number of new constants increases to nine.

The history of microstructure theories goes back to works by W.Voigt, who was the first to introduce a model of the medium with rotational interaction of its particles for studying elastic properties of a crystal. An early effort to develop an elasticity theory with asymmetric stress tensor evidently belongs to E.Cosserat and F.Cosserat. According to the Cosserat brothers' conception, which takes into account rotational interactions of material particles the most effective approach to the problems of stress-strain state in deformable solids is to introduce in the problem formulation the couple-stresses (moment of force per unit of area) in addition to the ordinary stresses (force per unit of area).

There has been a number of works reported in the literature in which the asymmetric theory is extended to the case of thermoelasticity and large deformations. Few works presenting solutions to a number of dynamic problems are also available in the literature. This is, for example, a systematic development of the modern theory by V.I.Erofeev, who considers the problem of propagation and interaction of elastic waves in solids with microstructure. Moreover, the idea of allowing for the internal rotation vector is often used for modeling plastic deformation in materials. However, a detailed discussion of these problems is beyond the scope of this paper, which is restricted to a static state of plane bodies in the framework of the elastic Cosserat continuum theory.

The asymmetric theory of elasticity for the Cosserat continuum (especially for the pseudo-Cosserat continuum) was successfully used by many authors to construct exact analytical solutions. In the majority works the obtained solutions are analyzed and compared with the corresponding solutions of the classical elasticity theory. In this comparison, new physical constants specifying the contribution of the couple-stress components generally assume the values from the energetically admissible range. This can be explained by deficiency of information on the material constants of microstructure media, which is one of the main factors restricting further investigation of asymmetric media models.

In some works a comparison between the solutions of the asymmetric and classical theories is carried out based on the analysis of the stress concentration coefficient and its dependence on the characteristic dimension of the stress concentrator. The analysis clearly demonstrated that compared to the classical theory the coefficient of the stress concentration increases (or decreases) with characteristic dimension of the concentrator. Although this fact is of obvious interest, the use of the concentration coefficient as a measurable parameter seems to be rather problematic. Thus, for example, an attempt to measure variation of the concentration index by the photoelasticity method has failed, since the resolving power of this method is too low to apply strictly to the desired characteristic dimension of the concentrator.

Considerable efforts have been spent on the development of analytical solutions to bending and torsion problems for rods with different cross-sections in terms of the asymmetric elasticity theory. In these studies a comparison of the couple-stress and classical solutions was based on the analysis of dependence of flexural and torsion stiffness on the characteristic dimension. Indeed, in the sense of experimental realization, stiffness is a well-measured parameter. However it is unlikely that the experimental measurements of flexural (torsion) stiffness can demonstrate the couple-stress response of the medium. This is due to the fact that such problems are missing one of the necessary conditions for the medium to show the couple-stress effect, namely a high stress gradient.

Therefore the above approaches if viewed as the examples of the couple-stress response of the medium are thought to be unrealizable from the viewpoint of their experimental implementation.

The objectives of the present project are as follows: to develop and analyze exact analytical solutions to a number of one-dimensional and two-dimensional static boundary-value problems in the framework of elastic Cosserat theory; to identify, based on the obtained solutions, measurable macro-quantities carrying information on "the couple" response of the examined material; to determine and compare the degree of difference between the introduced macro-quantities for the Cosserat, pseudo-Cosserat, and classical continua; to select problems that are most informative from the viewpoint of couple response effects, all other physical parameters being equal.

(a) Shear deformation of a plane infinite layer (plate) fixed at both edges under the action of gravitational force. Despite its simplicity this problem is used in this work as an example of the stress-strain state at a low stress gradient at which the medium is expected to exhibit couple stress properties only slightly.

(b) Torsion deformation of a ring rigidly fixed at the external contour due to rotation of its internal contour by a prescribed angle. Like the following problems, this case is characterized by a significant stress gradient compared to the first problem.

(c) Deformation of a ring rigidly fixed at the external contour due to displacement of the internal contour by a prescribed value.

(d) he Kirsch problem on uniaxial tension of an infinite plate loosened in the center by a circular hole. The extension of this problem to the pseudo-Cosserat continuum can be found in the work of R.D.Mindlin. V.A.Palmov determined the stress concentration in the vicinity of the circular hole in the context of the Cosserat continuum. It should be noted that the solution presented by V.A.Palmov does not allow us to analyze in full measure the stress-strain state in the vicinity of the hole, in particular, to estimate the degree of hole distortion under deformation.

(e) Problem on deformation of body of revolution.

The solutions to the above problems are exact and are represented in dimensionless form using the Bessel function of various orders.

[1]	M.A.Kulesh, V.P.Matveenko and I.N.Shardakov Construction of the analytical solution of the two-dimensional problem of the asymmetrical elasticity theory (in Russian) // PSTU Bulletin. Computational Mathematics and Mechanics. P. 55-60 (2000). Full text (pdf, rus, 4234Kb)
[2]	M.A.Kulesh Construction of exact analytical solutions to some two-dimensional couple-stress elastic problems and their parametrical analysis (in Russian) // Book of abstracts of Regional Conference "Youth science of Perm" (2000, Perm). No. 1. P. 148. Full text (pdf, rus, 1300Kb)
[3]	I.N.Shardakov and M.A.Kulesh Construction and analysis of some exact analytical solutions to two-dimensional elastic problems in the context of the Cosserat continuum (in Russian) // PSTU Bulletin. Mathematical simulation of systems and processes. No. 9. P. 187-201 (2001). Full text (pdf, rus, 2540Kb)
[4]	M.A.Kulesh and I.N.Shardakov Construction and analysis of analytical solutions to some one- and two-dimensional couple-stress elastic problems (in Russian) // Book of abstracts of VIII All-Russian Congress on Theoretical and Applied Mechanics (23-29 August 2001, Perm). P. 380. Full text (pdf, rus, 438Kb); About the conference
[5]	Kulesh M.A. and Shardakov I.N Construction and analysis of some exact analytical solutions to two-dimensional elastic problems in the framework of Cosserat continuum (in Russian) // Proceedings and abstracts of Republican Scientific-Technical Conference "Modern Problems of Mechanical Engineering" (29-31 October 2001, Tashkent-Samarkand). P. 232-236. Full text (pdf, rus, 1842Kb)
[6]	Kulesh M.A Construction and analysis of analytical solutions to some two-dimensional static problems in the framework of the asymmetrical elasticity theory (in Russian) // Synopsis of the dissertation for Ph.D degree in physics and mathematics. Perm, ICMM UB RAS (2001). Full text (pdf, rus, 909Kb); Synopsis (pdf, rus, 256Kb)
[7]	M.A.Kulesh, V.P.Matveenko and I.N.Shardakov Exact analytical solution of the Kirsch problem within the framework of the Cosserat continuum and pseudocontinuum // Journal of Applied Mechanics and Technical Physics. V. 42. No. 4. P. 145-154 (2001). Full text (pdf, rus, 1149Kb); Abstract (html, rus); About the journal
[8]	M.A.Kulesh Construction and analysis of analytical solutions to some static problems of the nonsymmetrical elasticity theory (in Russian) // Book of abstracts of Young Scientists Conference "Non-equilibrium transitions in continuum" (13-14 December 2002, Perm). P. 75. Full text (pdf, rus, 777Kb)
[9]	M.A.Kulesh, V.P.Matveenko and I.N.Shardakov Construction of analytical solutions of some two-dimensional problems of the couple-stress elasticity theory (in Russian) // Proceedings of RAS, Solid State Mechanics. No. 5. P. 69-82 (2002). Full text (pdf, rus, 6921Kb)
[10]	M.A.Kulesh, V.P.Matveenko and I.N.Shardakov Parametric analysis of analytical solutions to one- and two-dimensional problems in couple-stress theory of elasticity // Z. Angew. Math. Mech. V. 83. No. 4. P. 238-248 (2003). Full text (pdf, eng, 173Kb); Abstract (html, eng); About the journal
[11]	V.P.Matveenko, V.V.Korepanov, M.A.Kulesh, I.N.Shardakov Investigation of couple-stress effects in elastic bodies under deformation // ICTAM04 Abstracts Book and CD-ROM Proceedings, IPPT PAN, Warsaw, ISBN 83-89697-01-1 (2004). Full text (pdf, eng, 112Kb); About the conference